Show that if N lies in the array, then 2N + 1 is not prime.
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Find a general formula describing the elements of the array.
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First turn the statement "if N does not lie in the array, then 2N + 1 is prime" into its logical equivalent "if 2N + 1 is not prime, then N does lies in the array". It's this second statement we'll prove.
Since N lies in the array, it is equal to (2n+1)m+n for some n and m. Use this to work out 2N+1 in terms of m and n:
2N+1=2[(2n+1)m+n]+1=2(2n+1)m+ 2n+1=(2n+1)(2m+1).
The first row starts with the number 4. The starting number of any other row is 3 steps on from that of the previous one. This tells us that the first number in row m is of the form
Turn the statement "if N does not lie in the array, then 2N + 1 is prime" into its logical equivalent "if 2N + 1 is not prime, then N lies in the array".
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Numbers that are not prime can be factorised into two integers, both greater than 1.
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What is the first entry of the mth row in terms of m? What is the difference between successive numbers in the mth row in terms of m? And how do you express the nth entry of a row in terms of n and m?
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This may seem like an odd question — after all, he’s won — but it opens up some deep philosophical issues surrounding probability. David Spiegelhalter investigates how probability can be defined.
This may seem like an odd question — after all, he’s won — but it opens up some deep philosophical issues surrounding probability.
